Square Root of 1640

The square root is the inverse of the square of the number. 1640 is not a perfect square. The square root of 1640 is expressed in both radical and exponential form. In the radical form, it is expressed as √1640, whereas (1640)^(1/2) in the exponential form. √1640 ≈ 40.49753, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where the long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 1640 is broken down into its prime factors.

Step 1: Find the prime factors of 1640. Breaking it down, we get 2 x 2 x 2 x 5 x 41: 2^3 x 5^1 x 41^1

Step 2: Now, we found the prime factors of 1640. The second step is to make pairs of those prime factors. Since 1640 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 1640 using prime factorization is impossible.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: Begin by grouping the numbers from right to left. In the case of 1640, we need to group it as 40 and 16.

Step 2: Now, find n whose square is less than or equal to 16. We can say n as ‘4’ because 4 x 4 = 16. Now the quotient is 4, and after subtracting 16-16, the remainder is 0.

Step 3: Bring down 40, which is the new dividend. Add the old divisor with the same number 4 + 4 = 8, which will be our new divisor.

Step 4: The new divisor will be 8. We place a decimal point and add two zeros to the dividend, making it 4000.

Step 5: Next, 8n x n ≤ 4000. Let us consider n as 4, then 84 x 4 = 336.

Step 6: Subtract 336 from 400, the difference is 64, and the quotient is 40.

Step 7: The next dividend is 6400. Continue the steps until you achieve two decimal places in the quotient.

So the square root of √1640 ≈ 40.50

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 1640 using the approximation method.

Step 1: Find the closest perfect squares of √1640.

The smallest perfect square less than 1640 is 1600, and the largest perfect square greater than 1640 is 1681.

√1640 falls somewhere between 40 and 41.

Step 2: Apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

(1640 - 1600) / (1681 - 1600) = 40/81 ≈ 0.494

Using the formula, we identified the decimal point of our square root.

Add the whole number value to the decimal number: 40 + 0.494 = 40.494. Thus, the square root of 1640 is approximately 40.494.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because it is 4^2.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.
   
• Long division method: A method used to divide two numbers to find the quotient, which can also be adapted to find square roots of non-perfect squares through step-by-step division and approximation.